Containers employ lids of all shapes and sizes with various dispensing structures. For example, a hole or an array of holes is often provided in the lid for dispensing the contents of the container. A door is operable to cover and reveal these dispensing holes. As disclosed in U.S. Pat. Nos. 6,742,666 and 6,732,873, one type of door is a double hinged door. Such a door typically comprises two different sections hinged together. One of these door sections is hinged to the lid. The structure of the lid beneath the door is designed such that pressing downwardly on the door section hinged to the lid causes the other door section to flip upwardly revealing the hole or other dispensing structure. The present invention is directed toward improvements over the state of the art.